If you enroll your son into a school he needs a visa, discuss with the school to get the needed documents so you can apply for a Non Immigrant ED visa for the boy.
Best option to stay inside Thailand is to apply before you go for the visa that fits your purpose and length of stay.
If you need to work to gain income: The Non Immigrant B visa with work permit.
If you like to volunteer: the Non Immigrant O visa with work permit.
If you like to study for 1 year the Non Immigrant ED visa with documentation from the school or university.
If you like to retire, the Non Immigrant O visa
And I can go on and on and on, please do some research at the website of your local Thai Embassy to see the long list of long term visa possibilities and the Thai Immigration website for 1 year Extensions of Stay.
Only source who can tell you to be legally working without a work permit is the Ministry of Labor, the only source who can tell you to be legally working without a visa that allows you to work is the Immigration Office.
There are no probation periods, try out periods etc, it is well known that schools try to let people work without correct documents and when the ...... hits the fan, the employee is in trouble and the school just hire a new foreigner eager to make a few Thai baht during his/her holiday.
I have no idea on what reason the Non Immigrant O visa is issued, but if you have Thai family you may apply for the 60 days Extension of Stay to visit your Thai family otherwise it will be the 1 year Extension of Stay.
No, these double entry Tourist visa are not being sold for many years.
There is now a Multiple Entry Tourist Visa, valid for 6 months and on each entry during these 6 months you get 60 days of stay and each can be extended with another 30 days at local Immigration. With good use this visa can give almost 270 days inside Thailand.
If all document are uploaded correctly it take usually a few days to issue the visa by e-mail. The visa is valid for 3 months and you need to upload flight details.
You can change your flight once the visa is issued but be aware that you must use the visa during the three months of validity.
Sunday is the Immigration Office closed, so no problem to go on monday as first working day after closure. But it could be busy on mondays, best is to go friday and the Extension of Stay is added to your current stay.